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Multi-site, multivariate weather generator using maximum entropy bootstrap

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Abstract

Weather generators are increasingly becoming viable alternate models to assess the effects of future climate change scenarios on water resources systems. In this study, a new multisite, multivariate maximum entropy bootstrap weather generator (MEBWG) is proposed for generating daily weather variables, which has the ability to mimic both, spatial and temporal dependence structure in addition to other historical statistics. The maximum entropy bootstrap (MEB) involves two main steps: (1) random sampling from the empirical cumulative distribution function with endpoints selected to allow limited extrapolation and (2) reordering of the random series to respect the rank ordering of the original time series (temporal dependence structure). To capture the multi-collinear structure between the weather variables and between the sites, we combine orthogonal linear transformation with MEB. Daily weather data, which include precipitation, maximum temperature and minimum temperature from 27 years of record from the Upper Thames River Basin in Ontario, Canada, are used to analyze the ability of MEBWG based weather generator. Results indicate that the statistics from the synthetic replicates were not significantly different from the observed data and the model is able to preserve the 27 CLIMDEX indices very well. The MEBWG model shows better performance in terms of extrapolation and computational efficiency when compared to multisite, multivariate K-nearest neighbour model.

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Acknowledgments

The authors would like to acknowledge the financial support by the Natural Sciences and Engineering Research Council of Canada provided through the Discovery Grant to the second author. The authors would also like to thank Environment Canada for providing the climate data used in this research. We wish to acknowledge the effort put in by Prof. Zhu Li, the anonymous reviewer and the Executive Editor Jean-Claude Duplessy and thank them for their words of encouragement, good suggestions, and constructive comments.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Western University, London, ON, Canada

    Roshan K. Srivastav & Slobodan P. Simonovic

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  1. Roshan K. Srivastav
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  2. Slobodan P. Simonovic
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Correspondence to Roshan K. Srivastav.

Appendix

Appendix

The following procedure presents a hypothetical example that illustrates the use of maximum entropy bootstrap (MEB) for generating a single replicate from the original time series. Table 3 and Fig. 16 provide details of the example in which rt and rgt represent original and generated series for a time period t = 1, 2, 3, …, 7. The steps in Table 1 are explained below.

Table 3 Hypothetical example for generating replicates using MEB
Full size table
Fig. 16
figure 16

Hypothetical example: a empirical distribution of rt and b original and generated series preserving time dependent structure

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  1. 1.

    Sort the original time series (Table 3—Col. 2) in ascending order (Col. 4).

  2. 2.

    Calculate the intermediate points, which is average between the consecutive values in Column 5 and are shown in Column 6.

  3. 3.

    The upper (UL) and lower (LL) limiting values are calculates using Eqs. 9–11. Assuming 10 % trimming, we obtain LL = −3.67 and UL = 32.67. The uniform density is the combination of LL, intermediate points and UL as shown in Fig. 15.

  4. 4.

    Calculate the desired means (Col. 6), using Eqs. 12–14, which ensures they satisfy the ergodic theorem.

  5. 5.

    The quantiles of the maximum entropy density are computed by generate uniform random numbers between 0 and 1. This is shown in Column 7.

  6. 6.

    Calculate the new values using maximum entropy density, as shown in Column 8.

  7. 7.

    The replicate can be obtained by sorting Column 8 using the sorted order from Column 3.

It is evdient from the Fig. 16 that the MEB generated replicate is able to mimic the data pattern very well and hence the temporal and other statistics are well preserved.

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Srivastav, R.K., Simonovic, S.P. Multi-site, multivariate weather generator using maximum entropy bootstrap. Clim Dyn 44, 3431–3448 (2015). https://doi.org/10.1007/s00382-014-2157-x

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